Estimating a Random Walk First-Passage Time from Noisy or Delayed Observations

TL;DR

This paper investigates optimal methods for estimating the first passage time of a random walk observed with noise or delay, providing theoretical bounds and asymptotically optimal stopping rules validated by simulations.

Contribution

It introduces new bounds and optimal stopping rules for estimating first-passage times under noisy or delayed observations, extending existing theory to these challenging observation models.

Findings

Optimal stopping rules characterized for no-drift case.

Bounds established for drifted case in large-level regimes.

Asymptotically optimal stopping rule for noisy observations.

Abstract

A random walk (or a Wiener process), possibly with drift, is observed in a noisy or delayed fashion. The problem considered in this paper is to estimate the first time \tau the random walk reaches a given level. Specifically, the p-moment (p\geq 1) optimization problem \inf_\eta \ex|\eta-\tau|^p is investigated where the infimum is taken over the set of stopping times that are defined on the observation process. When there is no drift, optimal stopping rules are characterized for both types of observations. When there is a drift, upper and lower bounds on \inf_\eta \ex|\eta-\tau|^p are established for both types of observations. The bounds are tight in the large-level regime for noisy observations and in the large-level-large-delay regime for delayed observations. Noteworthy, for noisy observations there exists an asymptotically optimal stopping rule that is a function of a single observation. Simulation results are provided that corroborate the validity of the results for non-asymptotic settings.